I am scheduled to teach an upper-division undergraduate class on "Geometry" and I get to choose more or less what that means. Common choices seem to be non-Euclidean, hyperbolic, projective, or Erlangen geometry.
I would have liked to do differential geometry, since it seems to me to be a more central part of a mathematics education, but right now I think that that would probably be too difficult. In particular, linear algebra is not a prerequisite for the class. Also several of the students are future high-school teachers and would probably appreciate something that feels more relevant to them.
So I am wondering whether there is any textbook on non-Euclidean/hyperbolic/projective geometry (any or all) which at least leads up towards a "modern" differential-geometric point of view, e.g. the importance of a local metric and perhaps the intrinsic curvature. All the textbooks I have looked at so far seem to emphasize either the Euclid/Hilbert axiomatic style or the Klein Erlangen/transformation style of geometry, which are all well and good for the traditional (mostly homogeneous) examples; but I would like to at least expose the students to some of the ideas that lead later on to things like manifolds (and which are necessary for applications like relativity theory).